Ho!
In the not-too-distant past, I was playing a game of Warhammer 40k. All of the names, dates and armies have been changed in the interest of protecting the identities of those involved.

I will set the table: Warhammer event. Medium of size, medium in competitiveness. I was playing Chaos as I usually do, playing against a player playing Space Wolves. The game had begun but things had quickly taken a turn for the worse for my opponent. This was a critical turn (the bottom of turn 3) and my Great Unclean One was locked in combat with some Wolf Guard. I had slew a few on my turn 3, and a few more on his turn 3. It was time for his leadership check for losing combat with a minus two modifier. And he failed it. The combat was situated near his long table edge and it became very clear that if he broke and made it, he would run off of the board before ATSKNF kicked in and turned him around. Three inches isn't very far. No worries about not falling back though because I am Slow and Purposeful, so I couldn't sweep anyhow. He rolled a five.
Five inches is clearly off of the board. Clearly. My opponent looked carefully at the situation, look at his models, and chose to start to move his models 5 inches towards the short table edge on the left. I took note and corrected him.
"Oh!" he said, feigning surprise.
 He put his models back in roughly the same place... then moved everyone quickly to the long table edge. He clearly should have run off of the board.
I stopped him again and asked him why he's not moving off of the board.
"And they shall know no fear doesn't let them run off of the board." That, my friends, is also incorrect. I pulled out *his* book and showed him.
"Oh!" he said, feigning surprise.
Again, he put his models roughly back in their original position. It doesn't end here. Then began to measure from the model furthest from the long table edge, wrapped his tape around my Great Unclean One and stopped right at the table edge, which looked right about 5" give or take a hair. Then he began to snake his OTHER models AROUND my Great Unclean One to follow that original guy so they all stopped short. I stopped him again and explained that he could not do that, as he clearly had closer models to the board edge than the guy he started with and that he could not double back to go AROUND my guy to increase the chances they won't fall off of the board.
"Oh!" he said, feigning surprise.
As you can tell, I had an absolute blast playing this game. It was just a romp and so much fun was had by both parties.
...
I caught that guy trying to cheat THREE different times during the SAME event in the game. What were his motivations?
1. He knew he was losing the game and needed every advantage he could get.
2. He didn't want to lose the tournament.
3. He figured he knew the rules better than I did and thought he could get away with it.
4. He figured I wouldn't care if he bended the rules a bit to keep the game a little more competitive.
5. He is a truly evil individual.
6. Some combination of some or all of these things.

Of course, it's most likely #6. I can't get inside my opponent's head (any more than I usually do while playing) and figure out his exact motivation, but I can take a stab at it based on what I know about the circumstances concerning the situation.

All of us are motivated in different ways but for the most part, we are guided by our conscience to determine if we should act on those motivations. Lawrence Kohlberg, a psychologist, studied and wrote about the stages of moral development and boiled it down to three simple phases of reasoning:
(man, I love lists today!)
1. Pre-conventional reasoning. This stage is based on obedience and punishment. It is very self centered and self serving. Think young children. They behave in such a way to avoid punishment and obtain their desires. This type of reasoning allows for some seriously terrible things to occur, like lying, cheating, stealing and generally being a dick.
2. Conventional reasoning. This stage is based on social norms and peer to peer regulation. Most reasoning at this stage involved understanding what society/friends/family expect of you and you acting accordingly. Laws and authority regulate the behaviors that are deemed acceptable and not acceptable and it becomes the guiding light for the conventional thinker.
3. Post-conventional reasoning. Simply put, this is the self guided principle that there is a right and a wrong. That idea was developed through experience and there is an understanding that there are universal truths that transcend social norms, laws and authority. Slavery was a law, right?

There is a general understanding that we evolve through these stages throughout our lives. In theory, we are in pre-conventional as kids, conventional as teens/young adults and post-conventional as middle and late adults. The sad fact is, and I'm sure you can agree, that I know plenty of "adults" still stuck in pre-conventional reasoning. And sadly, I know quite a few tabletop gamers stuck here also.
Have you played a board game with a young child recently? Like Candy Land? Did you notice that all children expect to win? Like it never dawned on them that they won't win every game.

So what happened in my above example? Since we don't have a fully developed background on the day, or his previous games, or his family make up, or his mental prowess, or his stage in reasoning... etc., etc., etc., we have to make some assumptions based on what TYPE of cheating he was doing. Oh, another list! But this one isn't some hoity-toity list written by a psychologist. It's written by ME!

1. The Mechanical Cheat or PRE-Cheat. Example: Weighting dice, writing false army lists, modeling for advantage, changing your tape measure (haven't thought of this until I just typed it!). This type of cheating is done before hand and is pre-planned. It involves changing things that have a direct effect on the game but wouldn't be apparent in game. During the game, you'd still be civil and play within the rules, so an accusation of cheating wouldn't come up unless someone found the evidence of such during the game.
2. Cheating by Omission. Example: Your IC is SMOTE by a piece of wargear that causes instant death... nice to know, right? Cheating by omission is probably the most common form of cheating and is even done by players who generally have no intention of cheating. This is where you purposely allow your opponents to play outside of the rules, as long as it is harming them and not you. Additionally, it is purposely concealing important aspects of your army/list/game that are difficult/impossible to know unless YOU tell your opponent. Lastly, it is purposely not bringing up the rules that harm you from your own book/army, hiding behind the idea that your opponent is supposed to hold you accountable.
3. Cheating by Force of Will. Example: "These guys are move 12." "They don't get cover." "That's two dice." "They have Runes of Warding." Those statements don't make much sense out of context but illustrate just that. Force of will is a concept that applies to games like Warhammer. If you want your games to take 32 hours, then you should look up every rule every time you need to use it (how far can I move? Check the book! How do I shoot a bolter? Check the book!). Since we don't check the book for EVERY rule, we fall back on previous knowledge. If there is a player with a little more experience than the other, he will become the "rules" for the game. Instead of checking the book over and over, the other player will differ to the "rules" player during the game. This puts the "rules" player in a highly advantageous spot that he, if he was so inclined, could abuse easily. When both players are about equal, there will be a Force of Will battle where each will jockey to be this player... leading to some sort of argument and repeated shouts of "Judge!!!" When the "rules" player decides to intentionally quote incorrect rules for advantage, we have a CHEATER!
4. Cheating by Sleight of Hand. Example: the player rolls dice and picks them up and begins moving models, not stating what he was doing or why. Measuring quickly and moving models a little further for advantage. This is an easy cheat to catch and will demolish the credibility of the person doing it. It is my personal pet peeve in playing Warhammer and will ensure that I never play you again. I played against a player over the years that would routinely pick up and roll dice during all phases of the game. Two dice, three dice, one die, just pick em up and roll em. Low and behold when something was about to happen, he was still rolling dice. And suddenly he says "Two hits, two wounds." What? Who was shooting? At whom? When? I am disoriented!
5. I'm sure there are more but these seem to cover the basic concepts. Other examples could probably be jammed into one of those four categories.

My opponent was surely trying the Force of Will on me. He was attempting to use the thought that he was more well read in the rules than I was to effect the outcome of that combat in his favor. Unfortunately, he mistook my friendly and casual nature for aloofness.
I handled the situation appropriately I do believe. I was handily winning the game and this combat was going to solidify that point further. Perhaps he was motivated to cheat because he wanted to keep a sliver of hope alive but from my further interactions, it seemed that his cheating was done because he thought he could get away with it and have a better chance of winning if that interaction went his way.
We play a cooperative game, even though we love to pretend its competitive. The translation of Warhammer 40k into the competitive realm is often difficult. Instead of a fluent and smooth read, we get stuck with a Google-Translation rife with misspellings and nonsensical word pairings. Local FAQs, GW FAQs, addendums and supplements make for a tough me versus you game without clear cut rules and ways to implement those rules. There aren't judges umping your game like in baseball. You and your opponent need to AGREE to make things occur in the game. But cheating happens even in 'friendly' games.
And this is the big crux. If your opponent is cheating, simply don't agree with them. Continue to disagree until one of two things happens: The cheating stops or the game ends... and when left with a choice like that, I hope most people would choose to stop cheating and keep playing. Or you know, just RAGE QUIT!

What's the worst way someone has tried to cheat against you? How did you handle it?